Module Coq.Relations.Relation_Definitions

Section Relation_Definition.

   Variable A: Set.
   
   Definition relation := A -> A -> Prop.

   Variable R: relation.
   

Section General_Properties_of_Relations.

  Definition reflexive : Prop := (x: A) (R x x).
  Definition transitive : Prop := (x,y,z: A) (R x y) -> (R y z) -> (R x z).
  Definition symmetric : Prop := (x,y: A) (R x y) -> (R y x).
  Definition antisymmetric : Prop := (x,y: A) (R x y) -> (R y x) -> x=y.

  Definition equiv := reflexive /\ transitive /\ symmetric.

End General_Properties_of_Relations.

Section Sets_of_Relations.

   Record preorder : Prop := {
       preord_refl : reflexive;
       preord_trans : transitive }.

   Record order : Prop := {
       ord_refl : reflexive;
       ord_trans : transitive;
       ord_antisym : antisymmetric }.

   Record equivalence : Prop := {
       equiv_refl : reflexive;
       equiv_trans : transitive;
       equiv_sym : symmetric }.
   
   Record PER : Prop := {
       per_sym : symmetric;
       per_trans : transitive }.

End Sets_of_Relations.

Section Relations_of_Relations.

  Definition inclusion : relation -> relation -> Prop :=
      [R1,R2: relation] (x,y:A) (R1 x y) -> (R2 x y).

  Definition same_relation : relation -> relation -> Prop :=
      [R1,R2: relation] (inclusion R1 R2) /\ (inclusion R2 R1).
   
  Definition commut : relation -> relation -> Prop :=
      [R1,R2:relation] (x,y:A) (R1 y x) -> (z:A) (R2 z y)
                          -> (EX y':A |(R2 y' x) & (R1 z y')).

End Relations_of_Relations.

   
End Relation_Definition.

Hints Unfold reflexive transitive antisymmetric symmetric : sets v62.

Hints Resolve Build_preorder Build_order Build_equivalence
        Build_PER preord_refl preord_trans
        ord_refl ord_trans ord_antisym
        equiv_refl equiv_trans equiv_sym
        per_sym per_trans : sets v62.

Hints Unfold inclusion same_relation commut : sets v62.


Index